On 𝑡-Derivations of BCI-Algebras

Abstract

We introduce the notion of t-derivation of a BCI-algebra and investigate related properties. Moreover, we study t-derivations in a p-semisimple BCI-algebra and establish some results on t-derivations in a p-semisimple BCI-algebra.

1. Introduction

The notion of BCK-algebra was proposed by Imai and Iséki in 1966 [1]. In the same year, Iséki introduced the notion of a BCI-algebra [2], which is a generalization of a BCK-algebra. A series of interesting notions concerning BCI-algebras were introduced and studied, several papers have been written on various aspects of these algebras [3–5]. Recently, in the year 2004 [6], Jun and Xin have applied the notion of derivation in BCI-algebras which is defined in a way similar to the notion of derivation in rings and near-rings theory which was introduced by Posner in 1957 [7]. In fact, the notion of derivation in ring theory is quite old and plays a significant role in analysis, algebraic geometry and algebra.

After the work of Jun and Xin (2004) [6], many research articles have appeared on the derivations of BCI-algebras in different aspects as follows: in 2005 [8], Zhan and Liu have given the notion of 𝑓-derivation of BCI-algebras and studied 𝑝-semisimple BCI-algebras by using the idea of regular 𝑓-derivation in BCI-algebras. In 2006 [9], Abujabal and Al-Shehri have extended the results of BCI-algebras. Further, in the next year 2007 [10], they defined and studied the notion of left derivation of BCI-algebras and investigated some properties of left derivation in 𝑝-semisimple BCI-algebras. In 2009 [11], Öztürk and Çeven have defined the notion of derivation and generalized derivation determined by a derivation for a complicated subtraction algebra and discussed some related properties. Also, in 2009 [12], Öztürk et al. have introduced the notion of generalized derivation in BCI-algebras and established some results. Further, they have given the idea of torsion free BCI-algebra and explored some properties. In 2010 [13], Al-Shehri has applied the notion of left-right (resp., right-left) derivation in BCI-algebra to B-algebra and obtained some of its properties. In 2011 [14], Ilbira et al. have studied the notion of left-right (resp., right-left) symmetric biderivation in BCI-algebras.

Motivated by a lot of work done on derivations of BCI-algebras and on derivations of other related abstract algebraic structures, in this paper we introduce the notion of 𝑡-derivations on BCI-algebras and obtain some of its related properties. Further, we characterize the notion of 𝑝-semisimple BCI-algebra 𝑋 by using the notion of t-derivation and show that if 𝑑_𝑡 and 𝑑^′_𝑡 are 𝑡-derivations on 𝑋, then 𝑑_𝑡∘𝑑^′_𝑡 is also a 𝑡-derivation and 𝑑_𝑡∘𝑑^′_𝑡=𝑑^′_𝑡∘𝑑_𝑡. Finally, we prove that 𝑑_𝑡∗𝑑^′_𝑡=𝑑^′_𝑡∗𝑑_𝑡, where 𝑑_𝑡 and 𝑑^′_𝑡 are 𝑡-derivations on a 𝑝-semisimple BCI-algebra.

2. Preliminaries

We review some definitions and properties that will be useful in our results.

Definition 2.1 (see [2]). Let 𝑋 be a set with a binary operation “∗” and a constant 0. Then (𝑋,∗,0) is called a BCI algebra if the following axioms are satisfied for all 𝑥,𝑦,𝑧∈𝑋: (i)((𝑥∗𝑦)∗(𝑥∗𝑧))∗(𝑧∗𝑦)=0, (ii)(𝑥∗(𝑥∗𝑦))∗𝑦=0, (iii)𝑥∗𝑥=0, (iv)𝑥∗𝑦=0 and 𝑦∗𝑥=0⇒𝑥=𝑦.
Define a binary relation ≤ on 𝑋 by letting 𝑥∗𝑦=0 if and only if 𝑥≤𝑦. Then (𝑋,≤) is a partially ordered set. A BCI-algebra 𝑋 satisfying 0≤𝑥 for all 𝑥∈𝑋, is called BCK-algebra (see [1]).
In any BCI-algebra 𝑋 for all 𝑥,𝑦∈𝑋, the following properties hold. (1)(𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦. (2)𝑥∗0=𝑥. (3)(𝑥∗𝑧)∗(𝑦∗𝑧)≤𝑥∗𝑦. (4)𝑥∗0=0 implies 𝑥=0. (5)𝑥≤𝑦⇔𝑥∗𝑧≤𝑦∗𝑧 and 𝑧∗𝑦≤𝑧∗𝑥. A BCI-algebra 𝑋 is said to be associative if for all 𝑥,𝑦,𝑧∈𝑋, the following holds: (6)(𝑥∗𝑦)∗𝑧=𝑥∗(𝑦∗𝑧) [4]. Let 𝑋 be a BCI-algebra, we denote 𝑋₊={𝑥∈𝑋∣0≤𝑥}, the BCK-part of 𝑋 and by 𝐺(𝑋)={𝑥∈𝑋∣0∗𝑥=𝑥}, the BCI-G part of 𝑋. If 𝑋₊={0}, then 𝑋 is called a 𝑝-semisimple BCI-algebra. In a 𝑝-semisimple BCI-algebra 𝑋, the following properties hold.(7)𝑥∗(𝑥∗𝑦)=𝑦.(8)𝑥∗(0∗𝑦)=𝑦∗(0∗𝑥). (9)𝑥∗𝑦=0 implies 𝑥=𝑦. (10)(𝑥∗𝑧)∗(𝑦∗𝑧)=𝑥∗𝑦. (11)𝑥∗𝑎=𝑥∗𝑏 implies 𝑎=𝑏 that is left cancelable. (12)𝑎∗𝑥=𝑏∗𝑥 implies 𝑎=𝑏 that is right cancelable.

Definition 2.2 (see [6]). A subset 𝑆 of a BCI-algebra 𝑋 is called subalgebra of 𝑋 if 𝑥∗𝑦∈𝑆 whenever 𝑥,𝑦∈𝑆.
For a BCI-algebra 𝑋, we denote 𝑥∧𝑦=𝑦∗(𝑦∗𝑥) for all 𝑥,𝑦∈𝑋 [6]. For more details we refer to [3, 5, 6].

3. 𝑡-Derivations in a BCI-Algebra/𝑝-Semisimple BCI-Algebra

The following definitions introduce the notion of 𝑡-derivation for a BCI-algebra.

Definition 3.1. Let 𝑋 be a-BCI-algebra. Then for any 𝑡∈𝑋, we define a self map 𝑑_𝑡∶𝑋→𝑋 by 𝑑_𝑡(𝑥)=𝑥∗𝑡 for all 𝑥∈𝑋.

Definition 3.2. Let 𝑋 be a BCI-algebra. Then for any 𝑡∈𝑋, a self map 𝑑_𝑡∶𝑋→𝑋 is called a left-right 𝑡-derivation or (𝑙,𝑟)-𝑡-derivation of 𝑋 if it satisfies the identity 𝑑_𝑡(𝑥∗𝑦)=(𝑑_𝑡(𝑥)∗𝑦)∧(𝑥∗𝑑_𝑡(𝑦)) for all 𝑥,𝑦∈𝑋.
Similarly, we get the following.

Definition 3.3. Let 𝑋 be a BCI-algebra. Then for any 𝑡∈𝑋, a self map 𝑑_𝑡∶𝑋→𝑋 is called a right-left 𝑡-derivation or (𝑟,𝑙)-𝑡-derivation of 𝑋 if it satisfies the identity 𝑑_𝑡(𝑥∗𝑦)=(𝑥∗𝑑_𝑡(𝑦))∧(𝑑_𝑡(𝑥)∗𝑦) for all 𝑥,𝑦∈𝑋.
Moreover, if 𝑑_𝑡 is both a (𝑙,𝑟)- and a (𝑟,𝑙)-𝑡-derivation on 𝑋, we say that 𝑑_𝑡 is a 𝑡-derivation on 𝑋.

Example 3.4. Let 𝑋={0,1,2} be a BCI-algebra with the following Cayley table: ∗012010022102220 (3.1) For any 𝑡∈𝑋, define a self map 𝑑_𝑡∶𝑋→𝑋 by 𝑑_𝑡(𝑥)=𝑥∗𝑡 for all 𝑥∈𝑋. Then it is easily checked that 𝑑_𝑡 is a 𝑡-derivation of 𝑋.

Proposition 3.5. Let 𝑑_𝑡 be a self map of an associative BCI-algebra 𝑋. Then 𝑑_𝑡 is a (𝑙,𝑟)-𝑡-derivation of 𝑋.

Proof. Let 𝑋 be an associative BCI-algebra, then we have 𝑑_𝑡==𝑥∗𝑦𝑥∗𝑦∗𝑡𝑥∗𝑦∗𝑡∗0byProperty(6)and(2)=𝑥∗𝑦∗𝑡∗𝑥∗𝑦∗𝑡∗𝑥∗𝑦∗𝑡byProperty(iii)=𝑥∗𝑦∗𝑡∗𝑥∗𝑦∗𝑡∗𝑥∗𝑦∗𝑡byProperty(6)=𝑥∗𝑦∗𝑡∗𝑥∗𝑦∗𝑡∗(𝑥∗𝑡)∗𝑦byProperty(1)=()∧=𝑑𝑥∗𝑡∗𝑦𝑥∗𝑦∗𝑡_𝑡(𝑥)∗𝑦∧𝑥∗𝑑_𝑡𝑦. (3.2)

Proposition 3.6. Let 𝑑_𝑡 be a self map of an associative BCI-algebra 𝑋. Then, 𝑑_𝑡 is a (𝑟,𝑙)-𝑡-derivation of 𝑋.

Proof. Let 𝑋 be an associative BCI-algebra, then we have 𝑑_𝑡==)𝑥∗𝑦𝑥∗𝑦∗𝑡(𝑥∗𝑡∗𝑦∗0byProperty(1)and(2)=(𝑥∗𝑡)∗𝑦∗(𝑥∗𝑡)∗𝑦∗(𝑥∗𝑡)∗𝑦(as)=)∗)∗𝑥∗𝑥=0(𝑥∗𝑡∗𝑦(𝑥∗𝑡∗𝑦𝑥∗𝑦∗𝑡byProperty(1)=(𝑥∗𝑡)∗𝑦∗(𝑥∗𝑡)∗𝑦∗𝑥∗𝑦∗𝑡byProperty(6)=𝑥∗𝑦∗𝑡∧(𝑥∗𝑡)∗𝑦as=𝑦∗𝑦∗𝑥=𝑥∧𝑦𝑥∗𝑑_𝑡𝑦∧𝑑_𝑡(𝑥)∗𝑦. (3.3) Combining Propositions 3.5 and 3.6, we get the following Theorem.

Theorem 3.7. Let 𝑑_𝑡 be a self map of an associative BCI-algebra 𝑋. Then, 𝑑_𝑡 is a 𝑡-derivation of 𝑋.

Definition 3.8. A self map 𝑑_𝑡 of a BCI-algebra 𝑋 is said to be 𝑡-regular if 𝑑_𝑡(0)=0.

Example 3.9. Let 𝑋={0,𝑎,𝑏} be a BCI-algebra with the following Cayley table: ∗0𝑎𝑏0𝑎00𝑏𝑏𝑎0𝑏𝑏𝑏0 (3.4)
(i) For any 𝑡∈𝑋, define a self map 𝑑_𝑡∶𝑋→𝑋 by 𝑑_𝑡(𝑥)𝑏=𝑥∗𝑡=if0𝑥=0,𝑎if𝑥=𝑏. (3.5) Then it is easily checked that 𝑑_𝑡 is (𝑙,𝑟) and (𝑟,𝑙)-𝑡-derivations of 𝑋, which is not 𝑡-regular.
(ii) For any 𝑡∈𝑋, define a self map 𝑑^′_𝑡∶𝑋→𝑋 by 𝑑^′_𝑡(𝑥)0=𝑥∗𝑡=if𝑏𝑥=0,𝑎if𝑥=𝑏. (3.6) Then it is easily checked that 𝑑^′_𝑡 is (𝑙,𝑟) and (𝑟,𝑙)-𝑡-derivations of 𝑋, which is 𝑡-regular.

Proposition 3.10. Let 𝑑_𝑡 be a self map of a BCI-algebra 𝑋. Then (i)If 𝑑_𝑡 is a (𝑙,𝑟)-𝑡-derivation of 𝑋, then 𝑑_𝑡(𝑥)=𝑑_𝑡(𝑥)∧𝑥 for all 𝑥∈𝑋. (ii) If 𝑑_𝑡 is a (𝑟,𝑙)-𝑡-derivation of 𝑋, then 𝑑_𝑡(𝑥)=𝑥∧𝑑_𝑡(𝑥) for all 𝑥∈𝑋 if and only if 𝑑_𝑡 is t-regular.

Proof of (i). Let 𝑑_𝑡 be a (𝑙,𝑟)-𝑡-derivation of 𝑋, then 𝑑_𝑡(𝑥)=𝑑_𝑡()=𝑑𝑥∗0_𝑡(𝑥)∗0∧𝑥∗𝑑_𝑡(0)=𝑑_𝑡(𝑥)∧𝑥∗𝑑_𝑡(0=)𝑥∗𝑑_𝑡(0)∗𝑥∗𝑑_𝑡(0)∗𝑑_𝑡(𝑥=)𝑥∗𝑑_𝑡(0)∗𝑥∗𝑑_𝑡(𝑥)∗𝑑_𝑡(0)≤𝑥∗𝑥∗𝑑_𝑡(𝑥)byProperty(3)=𝑑_𝑡(𝑥)∧𝑥. (3.7) But 𝑑_𝑡(𝑥)∧𝑥≤𝑑_𝑡(𝑥) is trivial so (i) holds.

Proof of (ii). Let 𝑑_𝑡 be a (𝑟,𝑙)-𝑡-derivation of 𝑋. If 𝑑_𝑡(𝑥)=𝑥∧𝑑_𝑡(𝑥) then 𝑑_𝑡(0)=0∧𝑑_𝑡(0)=𝑑_𝑡(0)∗𝑑_𝑡(0)∗0=𝑑_𝑡(0)∗𝑑_𝑡(0)=0 (3.8) thereby implying 𝑑_𝑡 is 𝑡-regular. Conversely, suppose that 𝑑_𝑡 is 𝑡-regular, that is 𝑑_𝑡(0)=0, then we have 𝑑_𝑡(𝑥)=𝑑_𝑡()=𝑥∗0𝑥∗𝑑_𝑡(0)∧𝑑_𝑡(𝑥)=()∗0𝑥∗0∧𝑑_𝑡(𝑥)=𝑥∧𝑑_𝑡(𝑥). (3.9) This completes the proof.

Theorem 3.11. Let 𝑑_𝑡 be a (𝑙,𝑟)-𝑡-derivation of a 𝑝-semisimple BCI-algebra 𝑋. Then the following hold: (i)𝑑_𝑡(0)=𝑑_𝑡(𝑥)∗𝑥 for all 𝑥∈𝑋. (ii)𝑑_𝑡 is one-one. (iii)If 𝑑_𝑡 is t-regular, then it is an identity map. (iv)if there is an element 𝑥∈𝑋 such that 𝑑_𝑡(𝑥)=𝑥, then 𝑑_𝑡 is identity map. (v)if 𝑥≤𝑦, then 𝑑_𝑡(𝑥)≤𝑑_𝑡(𝑦) for all 𝑥,𝑦∈𝑋.

Proof of (i). Let 𝑑_𝑡 be a (𝑙,𝑟)-𝑡-derivation of a 𝑝-semisimple BCI-algebra 𝑋. Then for all 𝑥∈𝑋, we have 𝑥∗𝑥=0 and so 𝑑_𝑡(0)=𝑑_𝑡()=𝑑𝑥∗𝑥_𝑡(𝑥)∗𝑥∧𝑥∗𝑑_𝑡(𝑥)=𝑥∗𝑑_𝑡(𝑥)∗𝑥∗𝑑_𝑡(𝑥)∗𝑑_𝑡(𝑥)∗𝑥=𝑑_𝑡(𝑥)∗𝑥byproperty(7). (3.10)

Proof of (ii). Let 𝑑_𝑡(𝑥)=𝑑_𝑡(𝑦)⇒𝑥∗𝑡=𝑦∗𝑡, then by property (12), we have 𝑥=𝑦 and so 𝑑_𝑡 is one-one.

Proof of (iii). Let 𝑑_𝑡 be 𝑡-regular and 𝑥∈𝑋. Then, 0=𝑑_𝑡(0) so by the above part (i), we have 0=𝑑_𝑡(𝑥)∗𝑥 and hence by property (9), we obtain 𝑑_𝑡(𝑥)=𝑥 for all 𝑥∈𝑋. Therefore, 𝑑_𝑡 is the identity map.

Proof of (iv). It is trivial and follows from the above part (iii).

Proof of (v). Let 𝑥≤𝑦 implying 𝑥∗𝑦=0. Now, 𝑑_𝑡(𝑥)∗𝑑_𝑡𝑦=(𝑥∗𝑡)∗𝑦∗𝑡=𝑥∗𝑦byproperty(10)=0. (3.11) Therefore, 𝑑_𝑡(𝑥)≤𝑑_𝑡(𝑦). This completes the proof.

Definition 3.12. Let 𝑑_𝑡 be a 𝑡-derivation of a BCI-algebra 𝑋. Then, 𝑑_𝑡 is said to be an isotone 𝑡-derivation if 𝑥≤𝑦⇒𝑑_𝑡(𝑥)≤𝑑_𝑡(𝑦) for all 𝑥,𝑦∈𝑋.

Example 3.13. In Example 3.9(ii), 𝑑^′_𝑡 is an isotone 𝑡-derivation, while in Example 3.9(i), 𝑑_𝑡 is not an isotone 𝑡-derivation.

Proposition 3.14. Let 𝑋 be a BCI-algebra and 𝑑_𝑡 be a 𝑡-derivation on 𝑋. Then for all 𝑥,𝑦∈𝑋, the following hold: (i)If 𝑑_𝑡(𝑥∧𝑦)=𝑑_𝑡(𝑥)∧𝑑_𝑡(𝑦), then 𝑑_𝑡 is an isotone 𝑡-derivation. (ii)If 𝑑_𝑡(𝑥∗𝑦)=𝑑_𝑡(𝑥)∗𝑑_𝑡(𝑦), then 𝑑_𝑡 is an isotone 𝑡-derivation.

Proof of (i). Let 𝑑_𝑡(𝑥∧𝑦)=𝑑_𝑡(𝑥)∧𝑑_𝑡(𝑦). If 𝑥≤𝑦⇒𝑥∧𝑦=𝑥 for all 𝑥,𝑦∈𝑋. Therefore, we have 𝑑_𝑡(𝑥)=𝑑_𝑡𝑥∧𝑦=𝑑_𝑡(𝑥)∧𝑑_𝑡𝑦≤𝑑_𝑡𝑦. (3.12) Henceforth 𝑑_𝑡(𝑥)≤𝑑_𝑡(𝑦) which implies that 𝑑_𝑡 is an isotone 𝑡-derivation.

Proof of (ii). Let 𝑑_𝑡(𝑥∗𝑦)=𝑑_𝑡(𝑥)∗𝑑_𝑡(𝑦). If 𝑥≤𝑦⇒𝑥∗𝑦=0 for all 𝑥,𝑦∈𝑋. Therefore, we have 𝑑_𝑡(𝑥)=𝑑_𝑡(𝑥∗0)=𝑑_𝑡𝑥∗𝑥∗𝑦=𝑑_𝑡(𝑥)∗𝑑_𝑡𝑥∗𝑦=𝑑_𝑡(𝑥)∗𝑑_𝑡(𝑥)∗𝑑_𝑡𝑦≤𝑑_𝑡𝑦byproperty(ii). (3.13) Thus, 𝑑_𝑡(𝑥)≤𝑑_𝑡(𝑦). This completes the proof.

Theorem 3.15. Let 𝑑_𝑡 be a 𝑡-regular (𝑟,𝑙)-𝑡-derivation of a BCI-algebra 𝑋. Then, the following hold: (i)𝑑_𝑡(𝑥)≤𝑥 for all 𝑥∈𝑋. (ii)𝑑_𝑡(𝑥)∗𝑦≤𝑥∗𝑑_𝑡(𝑦) for all 𝑥,𝑦∈𝑋. (iii)𝑑_𝑡(𝑥∗𝑦)=𝑑_𝑡(𝑥)∗𝑦≤𝑑_𝑡(𝑥)∗𝑑_𝑡(𝑦) for all 𝑥,𝑦∈𝑋. (iv)ker(𝑑_𝑡)∶={𝑥∈𝑋∶𝑑_𝑡(𝑥)=0} is a subalgebra of 𝑋.

Proof of (i). For any 𝑥∈𝑋, we have 𝑑_𝑡(𝑥)=𝑑_𝑡(𝑥∗0)=(𝑥∗𝑑_𝑡(0))∧(𝑑_𝑡(𝑥)∗0)=(𝑥∗0)∧(𝑑_𝑡(𝑥)∗0)=𝑥∧𝑑_𝑡(𝑥)≤𝑥.

Proof of (ii). Since 𝑑_𝑡(𝑥)≤𝑥 for all 𝑥∈𝑋, then 𝑑_𝑡(𝑥)∗𝑦≤𝑥∗𝑦≤𝑥∗𝑑_𝑡(𝑦) and hence the proof follows.

Proof of (iii). For any 𝑥,𝑦∈𝑋, we have 𝑑_𝑡𝑥∗𝑦=𝑥∗𝑑_𝑡𝑦∧𝑑_𝑡(𝑥)=𝑑∗𝑦_𝑡(𝑥)∗𝑦∗𝑑_𝑡(𝑥)∗𝑦∗𝑥∗𝑑_𝑡𝑦=𝑑_𝑡(𝑥)∗𝑦∗0=𝑑_𝑡(𝑥)∗𝑦≤𝑑_𝑡(𝑥)∗𝑑_𝑡𝑦. (3.14)

Proof of (iv). Let 𝑥,𝑦∈ker(𝑑_𝑡)⇒𝑑_𝑡(𝑥)=0=𝑑_𝑡(𝑦). From (iii), we have 𝑑_𝑡(𝑥∗𝑦)≤𝑑_𝑡(𝑥)∗𝑑_𝑡(𝑦)=0∗0=0 implying 𝑑_𝑡(𝑥∗𝑦)≤0 and so 𝑑_𝑡(𝑥∗𝑦)=0. Therefore, 𝑥∗𝑦∈ker(𝑑_𝑡). Consequently ker(𝑑_𝑡) is a subalgebra of 𝑋. This completes the proof.

Definition 3.16. Let 𝑋 be a BCI-algebra and let 𝑑_𝑡, 𝑑^′_𝑡 be two self maps of 𝑋. Then we define 𝑑_𝑡∘𝑑^′_𝑡∶𝑋→𝑋 by (𝑑_𝑡∘𝑑^′_𝑡)(𝑥)=𝑑_𝑡(𝑑^′_𝑡(𝑥)) for all 𝑥∈𝑋.

Example 3.17. Let 𝑋={0,𝑎,𝑏} be a BCI algebra which is given in Example 3.4. Let 𝑑_𝑡 and 𝑑^′_𝑡 be two self maps on 𝑋 as defined in Example 3.9(i) and Example 3.9(ii), respectively.
Now, define a self map 𝑑_𝑡∘𝑑^′_𝑡∶𝑋→𝑋 by 𝑑_𝑡∘𝑑^′_𝑡𝑥)=0(if𝑏𝑥=𝑎,𝑏if𝑥=0. (3.15) Then, it is easily checked that (𝑑_𝑡∘𝑑^′_𝑡)(𝑥)=𝑑_𝑡(𝑑^′_𝑡(𝑥)) for all 𝑥∈𝑋.

Proposition 3.18. Let 𝑋 be a 𝑝-semisimple BCI-algebra 𝑋 and let 𝑑_𝑡, 𝑑^′_𝑡 be (𝑙,𝑟)-𝑡-derivations of 𝑋. Then, 𝑑_𝑡∘𝑑^′_𝑡 is also a (𝑙,𝑟)-𝑡-derivation of 𝑋.

Proof. Let 𝑋 be a 𝑝-semisimple BCI-algebra. 𝑑_𝑡 and 𝑑^′_𝑡 are (𝑙,𝑟)-𝑡-derivations of 𝑋. Then for all 𝑥,𝑦∈𝑋, we get 𝑑_𝑡∘𝑑^′_𝑡𝑥∗𝑦=𝑑_𝑡𝑑^′_𝑡𝑥∗𝑦=𝑑_𝑡𝑑^′_𝑡(𝑥)∗𝑦∧𝑥∗𝑑^′_𝑡𝑦=𝑑_𝑡𝑥∗𝑑^′_𝑡𝑦∗𝑥∗𝑑^′_𝑡𝑦∗𝑑^′_𝑡(𝑥)∗𝑦=𝑑_𝑡𝑑^′_𝑡(𝑥)∗𝑦byproperty(7)=𝑥∗𝑑_𝑡𝑑^′_𝑡𝑦∗𝑥∗𝑑_𝑡𝑑^′_𝑡𝑦∗𝑑_𝑡𝑑^′_𝑡(𝑥)=𝑑∗𝑦_𝑡𝑑^′_𝑡(𝑥)∗𝑦∧𝑥∗𝑑_𝑡𝑑^′_𝑡𝑦=𝑑_𝑡∘𝑑^′_𝑡(𝑥)∗𝑦∧𝑥∗𝑑_𝑡∘𝑑^′_𝑡𝑦. (3.16) Therefore, (𝑑_𝑡∘𝑑^′_𝑡) is a (𝑙,𝑟)-𝑡-derivation of 𝑋.
Similarly, we can prove the following.

Proposition 3.19. Let 𝑋 be a 𝑝-semisimple BCI-algebra and let 𝑑_𝑡, 𝑑^′_𝑡 be (𝑟,𝑙)-𝑡-derivations of 𝑋. Then 𝑑_𝑡∘𝑑^′_𝑡 is also a (𝑟,𝑙)-𝑡-derivation of 𝑋.

Combining Propositions 3.18 and 3.19, we get the following.

Theorem 3.20. Let 𝑋 be a 𝑝-semisimple BCI-algebra and let 𝑑_𝑡, 𝑑^′_𝑡 be 𝑡-derivations of 𝑋. Then, 𝑑_𝑡∘𝑑^′_𝑡 is also a 𝑡-derivation of 𝑋.

Now, we prove the following theorem.

Theorem 3.21. Let 𝑋 be a 𝑝-semisimple BCI-algebra and let 𝑑_𝑡, 𝑑^′_𝑡 be 𝑡-derivations of 𝑋. Then 𝑑_𝑡∘𝑑^′_𝑡=𝑑^′_𝑡∘𝑑_𝑡.

Proof. Let 𝑋 be a 𝑝-semisimple BCI-algebra. 𝑑_𝑡 and 𝑑^′_𝑡, 𝑡-derivations of 𝑋. Suppose 𝑑^′_𝑡 is a (𝑙,𝑟)-𝑡-derivation, then for all 𝑥,𝑦∈𝑋, we have 𝑑_𝑡∘𝑑^′_𝑡𝑥∗𝑦=𝑑_𝑡𝑑^′_𝑡𝑥∗𝑦=𝑑_𝑡𝑑^′_𝑡(𝑥)∗𝑦∧𝑥∗𝑑^′_𝑡𝑦=𝑑_𝑡𝑥∗𝑑^′_𝑡𝑦∗𝑥∗𝑑^′_𝑡𝑦∗𝑑^′_𝑡(𝑥)∗𝑦=𝑑_𝑡𝑑^′_𝑡(𝑥)∗𝑦byproperty(7). (3.17) As 𝑑_𝑡 is a (𝑟,𝑙)-𝑡-derivation, then =𝑑^′_𝑡(𝑥)∗𝑑_𝑡𝑦∧𝑑_𝑡𝑑^′_𝑡(𝑥)∗𝑦=𝑑^′_𝑡(𝑥)∗𝑑_𝑡𝑦. (3.18) Again, if 𝑑_𝑡 is a (𝑟,𝑙)-𝑡-derivation, then we have 𝑑^′_𝑡∘𝑑_𝑡𝑥∗𝑦=𝑑^′_𝑡𝑑_𝑡𝑥∗𝑦=𝑑^′_𝑡𝑥∗𝑑_𝑡𝑦∧𝑑_𝑡(𝑥)∗𝑦=𝑑^′_𝑡𝑥∗𝑑_𝑡𝑦byproperty(7) (3.19) But 𝑑^′_𝑡 is a (𝑙,𝑟)-𝑡-derivation, then =𝑑^′_𝑡(𝑥)∗𝑑_𝑡𝑦∧𝑥∗𝑑^′_𝑡𝑑_𝑡𝑦=𝑑^′_𝑡(𝑥)∗𝑑_𝑡𝑦. (3.20) Therefore from (3.18) and (3.20), we obtain 𝑑_𝑡∘𝑑^′_𝑡𝑥∗𝑦=𝑑^′_𝑡∘𝑑_𝑡𝑥∗𝑦. (3.21) By putting 𝑦=0, we get 𝑑_𝑡∘𝑑^′_𝑡(𝑥)=𝑑^′_𝑡∘𝑑_𝑡(𝑥)∀𝑥∈𝑋. (3.22) Hence, 𝑑_𝑡∘𝑑^′_𝑡=𝑑^′_𝑡∘𝑑_𝑡. This completes the proof.

Definition 3.22. Let 𝑋 be a BCI-algebra and let 𝑑_𝑡, 𝑑^′_𝑡 be two self maps of 𝑋. Then we define 𝑑_𝑡∗𝑑^′_𝑡∶𝑋→𝑋 by (𝑑_𝑡∗𝑑^′_𝑡)(𝑥)=𝑑_𝑡(𝑥)∗𝑑^′_𝑡(𝑥) for all 𝑥∈𝑋.

Example 3.23. Let 𝑋={0,𝑎,𝑏} be a BCI algebra which is given in Example 3.4. Let 𝑑_𝑡 and 𝑑^′_𝑡 be two self maps on 𝑋 as defined in Example 3.9(i) and Example 3.9(ii), respectively.
Now, define a self map 𝑑_𝑡∗𝑑^′_𝑡∶𝑋→𝑋 by 𝑑_𝑡∗𝑑^′_𝑡𝑥)=0(if𝑏𝑥=𝑎,𝑏if𝑥=0. (3.23) Then, it is easily checked that (𝑑_𝑡∗𝑑^′_𝑡)(𝑥)=𝑑_𝑡(𝑥)∗𝑑^′_𝑡(𝑥) for all 𝑥∈𝑋.

Theorem 3.24. Let 𝑋 be a 𝑝-semisimple BCI-algebra and let 𝑑_𝑡, 𝑑^′_𝑡 be 𝑡-derivations of 𝑋. Then 𝑑_𝑡∗𝑑^′_𝑡=𝑑^′_𝑡∗𝑑_𝑡.

Proof. Let 𝑋 be a 𝑝-semisimple BCI-algebra. 𝑑_𝑡 and 𝑑^′_𝑡, 𝑡-derivations of 𝑋.
Since 𝑑^′_𝑡 is a (𝑟,𝑙)-𝑡-derivation of 𝑋, then for all 𝑥,𝑦∈𝑋, we have 𝑑_𝑡∘𝑑^′_𝑡𝑥∗𝑦=𝑑_𝑡𝑑^′_𝑡𝑥∗𝑦=𝑑_𝑡𝑥∗𝑑^′_𝑡𝑦∧𝑑^′_𝑡(𝑥)∗𝑦=𝑑_𝑡𝑥∗𝑑^′_𝑡𝑦byproperty(7). (3.24) But 𝑑_𝑡 is a (𝑙,𝑟)-𝑡-derivation, so =𝑑_𝑡(𝑥)∗𝑑^′_𝑡𝑦∧𝑥∗𝑑_𝑡𝑑^′_𝑡𝑦=𝑑_𝑡(𝑥)∗𝑑^′_𝑡𝑦. (3.25) Again, if 𝑑^′_𝑡 is a (𝑙,𝑟)-𝑡-derivation of 𝑋, then for all 𝑥,𝑦∈𝑋, we have 𝑑_𝑡∘𝑑^′_𝑡𝑥∗𝑦=𝑑_𝑡𝑑^′_𝑡𝑥∗𝑦=𝑑_𝑡𝑑^′_𝑡(𝑥)∗𝑦∧𝑥∗𝑑^′_𝑡𝑦=𝑑_𝑡𝑥∗𝑑^′_𝑡𝑦∗𝑥∗𝑑^′_𝑡𝑦∗𝑑^′_𝑡(𝑥)∗𝑦=𝑑_𝑡𝑑^′_𝑡(𝑥)∗𝑦byproperty(7). (3.26) As 𝑑_𝑡 is a (𝑟,𝑙)-𝑡-derivation, then =𝑑^′_𝑡(𝑥)∗𝑑_𝑡𝑦∧𝑑_𝑡𝑑^′_𝑡(𝑥)∗𝑦=𝑑^′_𝑡(𝑥)∗𝑑_𝑡𝑦. (3.27) Henceforth from (3.25) and (3.27), we conclude 𝑑_𝑡(𝑥)∗𝑑^′_𝑡𝑦=𝑑^′_𝑡(𝑥)∗𝑑_𝑡𝑦 (3.28) By putting 𝑦=𝑥, we get 𝑑_𝑡(𝑥)∗𝑑^′_𝑡(𝑥)=𝑑^′_𝑡(𝑥)∗𝑑_𝑡(𝑥)𝑑_𝑡∗𝑑^′_𝑡(𝑥)=𝑑^′_𝑡∗𝑑_𝑡(𝑥)∀𝑥∈𝑋. (3.29) Hence, 𝑑_𝑡∗𝑑^′_𝑡=𝑑^′_𝑡∗𝑑_𝑡. This completes the proof.

4. Conclusion

Derivation is a very interesting and important area of research in the theory of algebraic structures in mathematics. The theory of derivations of algebraic structures is a direct descendant of the development of classical Galois theory (namely, Suzuki [15] and Van der Put and Singer [16, 17]) and the theory of invariants. An extensive and deep theory has been developed for derivations in algebraic structures viz. BCI-algebras, 𝐶^∗-algebras, commutative Banach algebras and Galois theory of linear differential equations (see, e.g., Jun and Xin [6], Ara and Mathieu [18], Bonsall and Duncan [19], Murphy [20] and Villena [21] where further references can be found). It plays a significant role in functional analysis; algebraic geometry; algebra and linear differential equations.

In the present paper, we have considered the notion of 𝑡-derivations in BCI-algebras and investigated the useful properties of the 𝑡-derivations in BCI-algebras. Finally, we investigated the notion of 𝑡-derivations in a 𝑝-semisimple BCI-algebra and established some results on 𝑡-derivations in a 𝑝-semisimple BCI-algebra. In our opinion, these definitions and main results can be similarly extended to some other algebraic systems such as subtraction algebras [11], B-algebras [13], MV-algebras [22], d-algebras, Q-algebras and so forth. In future we can study the notion of 𝑡-derivations on various algebraic structures which may have a lot of applications in different branches of theoretical physics, engineering and computer science. It is our hope that this work would serve as a foundation for the further study in the theory of derivations of BCK/BCI-algebras.

In our future study of 𝑡-derivations in BCI-algebras, may be the following topics should be considered: (1)to find the generalized 𝑡-derivations of BCI-algebras, (2)to find more results in 𝑡-derivations of BCI-algebras and its applications, (3)to find the 𝑡-derivations of B-algebras, Q-algebras, subtraction algebras, d-algebra and so forth.

Acknowledgments

The authors would like to express their sincere thanks to the anonymous referees for their valuable comments and several useful suggestions. This research is supported by the Deanship of Scientific Research, University of Tabuk, Tabuk, Saudi Arabia.